$\Gamma$-limit of the cut functional on dense graph sequences

TL;DR

This paper investigates the asymptotic behavior of graph partitioning problems on dense graph sequences using graphons and $ ext{Gamma}$-convergence, providing a variational framework for community detection.

Contribution

It introduces a $ ext{Gamma}$-limit approach to the cut functional on dense graphs via graphons, linking graph partitioning to variational principles and Young measures.

Findings

The $ ext{Gamma}$-limit of the cut functional is characterized on graphons.

Partition problems can be expressed through probabilities of node community membership.

Insights into large graph bisection problems, which are NP-complete, are obtained.

Abstract

A sequence of graphs with diverging number of nodes is a dense graph sequence if the number of edges grows approximately as for complete graphs. To each such sequence a function, called graphon, can be associated, which contains information about the asymptotic behavior of the sequence. Here we show that the problem of subdividing a large graph in communities with a minimal amount of cuts can be approached in terms of graphons and the $\Gamma$-limit of the cut functional, and discuss the resulting variational principles on some examples. Since the limit cut functional is naturally defined on Young measures, in many instances the partition problem can be expressed in terms of the probability that a node belongs to one of the communities. Our approach can be used to obtain insights into the bisection problem for large graphs, which is known to be NP-complete.